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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>;<br/>

<b>assume </b><a NAME="E1:121_1_1"/><i><font color="Green" title="E78">A1</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="freealg.html#V3" title="FREEALG:attr.3">free</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">A'</font> = <font color="Maroon" title="c1">A</font> as  <a href="freealg.html#V3" title="FREEALG:attr.3">free</a> <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a> ;<br/>
<b>consider </b><font color="Maroon" title="c3">G</font> being  <a href="freealg.html#V2" title="FREEALG:attr.2">free</a>  <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">A'</font>;<br/>
<a NAME="E4:121_1_1"/><i><font color="Green" title="E79">A2</font></i>: 
 <a href="aofa_000.html#K10" title="AOFA_000:func.10">Generators</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><a class="ref" href="aofa_000.html#T34" target="_self" title="AOFA_000:th.34">Th34</a></i>;<br/>
<a NAME="E5:121_1_1"/>
 <a href="aofa_000.html#K10" title="AOFA_000:func.10">Generators</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="aofa_000.html#T71" target="_self" title="AOFA_000:th.71">Th71</a></i>;<br/>
<b>hence </b><a NAME="E6:121_1_1"/>
 <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c1">A</font> is    <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E4:121_1_1"><i><font color="Green" title="E79">A2</font></i></a>, <a class="ref" href="aofa_000.html#T31" target="_self" title="AOFA_000:th.31">Th31</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D25" target="_self" title="AOFA_000:def.25">AOFA_000:def 25</a><br/>

<b>thus </b><a NAME="E7:121_1_1"/>
for <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> ( not <!--HUMANRECFAILED-->( <font color="Olive" title="b1">I1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">I2</font> ) &amp; not <!--HUMANRECFAILED-->( <font color="Olive" title="b2">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I1</font> ) &amp; not <!--HUMANRECFAILED-->( ( <font color="Olive" title="b1">I1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> or <font color="Olive" title="b2">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> ) &amp; <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> ) )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:121_1_1"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="aofa_000.html#T72" target="_self" title="AOFA_000:th.72">Th72</a>, <a class="ref" href="aofa_000.html#T73" target="_self" title="AOFA_000:th.73">Th73</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D24" target="_self" title="AOFA_000:def.24">AOFA_000:def 24</a><br/>

<b>thus </b><a NAME="E8:121_1_1"/>
( ( for <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">I1</font>, <font color="Olive" title="b3">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  holds not  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">I1</font>,<font color="Olive" title="b3">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> ) &amp; ( for <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  holds not  <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> ) &amp; ( for <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font>, <font color="Olive" title="b3">C</font>, <font color="Olive" title="b4">J1</font>, <font color="Olive" title="b5">J2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  holds <br/>(  not <font color="Olive" title="b1">I1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> or  not <font color="Olive" title="b2">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> or  not <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Olive" title="b3">C</font>,<font color="Olive" title="b4">J1</font>,<font color="Olive" title="b5">J2</font> ) ) &amp; ( for <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font>, <font color="Olive" title="b3">C</font>, <font color="Olive" title="b4">J</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  holds <br/>(  not <font color="Olive" title="b1">I1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> or  not <font color="Olive" title="b2">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> or  not <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b3">C</font>,<font color="Olive" title="b4">J</font> ) ) &amp; ( for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">I1</font>, <font color="Olive" title="b3">I2</font>, <font color="Olive" title="b4">C2</font>, <font color="Olive" title="b5">J</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  holds not  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Olive" title="b1">C1</font>,<font color="Olive" title="b2">I1</font>,<font color="Olive" title="b3">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b4">C2</font>,<font color="Olive" title="b5">J</font> ) )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:121_1_1"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="aofa_000.html#T72" target="_self" title="AOFA_000:th.72">Th72</a>, <a class="ref" href="aofa_000.html#T73" target="_self" title="AOFA_000:th.73">Th73</a>, <a class="ref" href="aofa_000.html#T74" target="_self" title="AOFA_000:th.74">Th74</a></i>;<br/>


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